Coupled Fixed Point Theory over Quantale-Valued Quasi-Metric Spaces (QVQMS) with Applications in Generalized Metric Structures

Abstract

In this study, we establish several coupled fixed point results in quantale-valued quasi-metric spaces (QVQMSs), which constitutes a generalization of metric and probabilistic metric spaces. The obtained results will be illustrated with concrete examples. Furthermore, we introduce the concept of ${\theta }_s$-completeness and, as an application of the main theorems, we derive some results in both quantale-valued partial metric spaces and probabilistic metric spaces.

1. Introduction

Metric spaces serve as one of the cornerstone of topology and analysis, providing a framework for measuring distances between elements in a set. Extensions of this concept have been developed to address more general situations, such as quasi-metrics [1], partial metrics [2], and probabilistic metric spaces (PMSs) [3]. Quantale-valued generalized metric spaces, introduced by Flagg [4], also known as continuity spaces, are a significant generalization of metric spaces. This generalized metric space notion provides a unified setting that combines order theoretic and metric notions. Quantale-valued metric space is obtained by changing the value set $[0,\infty ]$ of a classical metric space with a quantale. Within the framework of quantale-valued generalized metric spaces, one can recover metric spaces and probabilistic metric spaces as special examples. Moreover, these structures play a significant role in quantitative domain theory and denotational semantics (see [5]). In [6], the notion of action was introduced, and a generalization of the Banach fixed point theorem was obtained in the setting of quantale-valued generalized metric spaces. In [7], the notion of an action was defined on the partially ordered semigroups, and fixed point results for Banach contraction-type mappings were established. For more detailed work on these spaces, see [8,9,10,11,12,13,14].

The Banach Contraction Principle [15] led to significant attention being given to fixed point theory (FPT), which is regarded as a powerful instrument in various diciplines. A variety of extensions of the Banach fixed point theorem have been developed, with the coupled fixed point theorem being a prominent example. This notion was initially presented by Bhaskar and Lakshmikantham [16] and was later expanded by Sabetghadam et al. [17] into the structure of complete cone metric spaces. In [18], Berinde obtained a fairly general and flexible extension of the results due to Bhaskar and Laksikantham [16]. It is worth noting that, in [18], the continuity of mapping is not required, and the contraction condition is more general than that of Bhaskar and Laksikantham [16]. In [19], Ćirić et al. generalized the results given in [16] to the partially ordered probabilistic metric spaces and established a number of new results. Recently, in [20], Çevik et al. extended Bhaskar and Lakshmikantham’s [16] results to the framework of ordered vector spaces.

While FPT has seen substantial progress across numerous generalized metric frameworks, research focusing on fixed point theorems within quantale-valued generalized metric spaces remains rather limited. In this paper, we investigate an appropriate generalization and extension of the coupled fixed point results established in [16,17,18], which can be obtained by transferring them to the framework of QVQMSs. Since coupled fixed point theorems had not been developed in QVQMSs, this topic represents a gap in the existing literature. We also examine which structural conditions are necessary to ensure a meaningful and effective adaptation of the techniques to the quantale-valued setting. As quantale-valued quasi-metric spaces constitute a common generalization of metric spaces and probabilistic metric spaces, the results obtained in this work yield, as special cases, several new coupled fixed point theorems in quasi-metric and probabilistic metric spaces. Moreover, coupled fixed point theorems have important applications in various areas of mathematics and economics (see [21]). Therefore, the coupled fixed point results established in this paper are expected to be of potential interest for applications in these fields.

We arrange the paper in the following manner. In the second section, we present some preliminary definitions and properties that will be frequently used throughout the paper. In the third section, inspired by the works of [7,17], we establish coupled fixed point results in quantale-valued quasi-metric spaces. Some of the results obtained [17] will be extended to this broader setting. In the final section, we extend the notion of 0-completeness introduced in [22] for quantale-valued partial metric spaces [23] and introduce the concept of ${\theta }_s$-completeness. Motivated by the study of [24], which reveals that fixed point results established in partial metric spaces can be obtained as consequences of those in metric spaces, we apply our main theorem from Section 2 to the quantale-valued partial metric space context. Moreover, since probabilistic metric spaces can be regarded as quantale-valued quasi-metric spaces, we end the paper by deriving coupled fixed point results for probabilistic metric spaces. The newly obtained fixed point theorems are expected to fill a gap in the literature and open perspectives for applications in denotational semantics and theoretical computer science.

2. Preliminaries

Let $(Q,\precsim )$ be a complete lattice, where every subset has a supremum and infimum. We will denote the bottom and top elements of the complete lattice Q by $\theta$ and e, respectively. The well above relation defined below is an extension of the strict inequality relation in [4], formulated in the context of complete lattices.

Definition 1 

(Well above relation, [4]). Let $(Q,\precsim )$ be a complete lattice. Let $p,q\in Q$. Then, p is said to be well above q, symbolized as $p\gg q$, if the following holds:

$$\mathrm{I}\mathrm{f} q\succsim infU,\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t} U\subseteq Q,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{o}\mathrm{n}\mathrm{e} \mathrm{c}\mathrm{a}\mathrm{n} \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d} u\in U \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} p\succsim u.$$

Moreover, if the following condition holds for all $p\in Q$, then Q is called a completely distributive lattice:

$$p=inf\{q\in Q:q\gg p\}.$$

Definition 2 

(Value quantale, [4]). A value quantale is defined as a triple $(Q,\precsim ,\oplus )$, satisfying the following requirements for all $\varrho \in Q$ and $\left\{q_i\right\}\subseteq Q$:

1. 

$(Q,\precsim )$ is a completely distributive lattice;

2. 

On Q, ⊕ serves as a binary operation satisfying associativity and commutativity;

3. 

$\varrho \oplus \theta =\varrho$;

4. 

$\varrho \oplus {inf}_{i\in I}{q}_i={inf}_{i\in I}(\varrho \oplus q_i)$, which ensures the infinite distributivity of the quantale operation ⊕ over infima;

5. 

$e\gg \theta$;

6. 

If ${\varrho }_1,{\varrho }_2\gg \theta$, then ${\varrho }_1\wedge {\varrho }_2\gg \theta$.

Let $\mu ,\nu \in Q$. The mapping $\dot{-}:Q\times Q\to Q$ is given by

$$\mu \dot{-}\nu =inf\{\lambda \in Q:\nu \oplus \lambda \succsim \mu \},$$

for further details on its properties, see [4].

Example 1 

([7]). Consider $Q=\left(\right[0,\infty ],\le ,+)$, where ≤ denotes the usual ordering and + denotes the usual addition on the extended real numbers. Then, Q is a value quantale.

Example 2 

([25]). Let $Q*=\{T,F\}$ be a two-point set ordered by $T\ge F$. If the operation ∨ is taken as the quantale operation, then $(Q*,\le ,\vee )$ is a value quantale. In particular, this quantale is referred to as the value quantale of truth values in [25].

Example 3. 

Let $Q^{\prime }$ be the set of all linear and nondecreasing functions $f:[0,\infty ]\to [0,\infty ]$ endowed with the pointwise order ≤. Then, $(Q^{\prime },\le )$ is a complete lattice whose bottom and top elements are the functions ${\zeta }_B$ and ${\zeta }_T$, respectively, defined by ${\zeta }_B\left(x\right)=0$ and ${\zeta }_T\left(x\right)=\infty$ for all $x\in [0,\infty ]$. Define an operator ⊕ on $Q^{\prime }$ by

$$(\zeta \oplus \rho )\left(x\right)=\zeta \left(x\right)+\rho \left(x\right)$$

for all $\zeta ,\rho \in Q^{\prime }$ and $x\in [0,\infty ]$. The operation ⊕ is commutative and associative. Moreover, $\zeta \oplus {\zeta }_B=\zeta$ for all $\zeta \in Q^{\prime }$. Let ${\left\{{\rho }_i\right\}}_{i\in I}\subseteq Q^{\prime }$. Then, for any $\zeta \in Q^{\prime }$, we have $\zeta \oplus inf\left\{{\rho }_i\right\}=inf\{\zeta \oplus {\rho }_i\}$. We now show that ${\zeta }_T\gg {\zeta }_B$. Let ${\left\{{\rho }_i\right\}}_{i\in I}\subseteq Q^{\prime }$ be such that ${\zeta }_B\ge inf{\left\{{\rho }_i\right\}}_{i\in I}$. Then, ${\zeta }_T\ge {\rho }_i$ for all $i\in I$, since ${\zeta }_T\left(x\right)=\infty$ for all $x\in [0,\infty ]$. Finally, condition 6 of the Definition 2 follows directly. Consequently, $(Q^{\prime },\le ,\oplus )$ is a value quantale.

Definition 3 

(Quantale-Valued Quasi Metric, see [4]). Consider a nonempty set S and value quantale $(Q,\precsim ,\oplus )$. A mapping $\delta :S\times S\to Q$ is referred to as quantale-valued quasi-metric whenever, for all $\varrho ,\upsilon ,\varsigma \in S$, the conditions below are satisfied:

$$\begin{array}{ccc}\hfill •& \delta (\varrho ,\varrho )=\theta ,\hfill & \hfill (\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{y})\\ \hfill •& \delta (\varrho ,\upsilon )=\delta (\upsilon ,\varrho )=\theta \Rightarrow \varrho =\upsilon ,\hfill & \hfill (\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{y})\\ \hfill •& \delta (\varrho ,\upsilon )\precsim \delta (\varrho ,\varsigma )\oplus \delta (\varsigma ,\upsilon ),\hfill & \hfill (\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{y})\end{array}$$

The pair $(S,\delta )$ is referred to as a quantale-valued quasi-metric space (also called a separated continuity space in [4]).

Consider a quantale-valued quasi-metric space $(S,\delta )$. Let $\sigma ,\upsilon \in S$. Then, the mappings ${\delta }^{-1}$ (dual of $\delta$) and ${\delta }^s$ (symmetrization of $\delta$) are specified, respectively, by

$${\delta }^{-1}(\sigma ,\upsilon )=\delta (\upsilon ,\sigma )\mathrm{and}{\delta }^s(\sigma ,\upsilon )=\delta (\sigma ,\upsilon )\vee \delta (\upsilon ,\sigma ),$$

see [7]. For the topological properties of this space, the reader may consult [4,6,7,25] for further details. Consider a net ${\left({\varrho }_{\lambda }\right)}_{\lambda \in \Lambda }$ in S. ${\left({\varrho }_{\lambda }\right)}_{\lambda \in \Lambda }$ is symmetrically convergent to $\varrho$ whenever, for any $\epsilon \gg \theta$, there exists ${\lambda }_0\in \Lambda$, such that for all $\lambda \succcurlyeq {\lambda }_0$, ${\delta }^s({\varrho }_{\lambda },\varrho )\precsim ϵ$. A net ${\left({\varrho }_{\lambda }\right)}_{\lambda \in \Lambda }$ will be termed Cauchy whenever, for every $\epsilon \gg \theta$, one can find ${\lambda }_0\in \Lambda$, such that $\delta ({\varrho }_{\alpha },{\varrho }_{\beta })\precsim ϵ$ holds for all indices $\alpha ,\beta \succcurlyeq {\lambda }_0$; that is, ${\delta }^s({\varrho }_{\alpha },{\varrho }_{\beta })\precsim ϵ$ (see [25]). Moreover, if every Cauchy net is symmetrically convergent, then S is called complete (see [25]). If we take a sequence instead of a net in the definition of completeness, then such a quantale-valued quasi-metric space is referred to as s-complete. Every complete quantale-valued quasi-metric space is necessarily s-complete.

Example 4 

([25]). Let us consider the value quantale $(Q,\precsim ,\oplus )$. The mapping $\delta :Q\times Q\to Q$ given by $\delta (\mu ,\nu )=\nu \dot{-}\mu$ is a quantale-valued quasi-metric on Q and $(Q,\delta )$ is s-complete (see Theorem 4.9 in [25]).

Example 5. 

Let $Q=\{e,\theta ,\alpha ,\beta \}$. Consider a partial order ≾ defined by

$$\precsim =\left\{\right(e,e),(\theta ,\theta ),(\alpha ,\alpha ),(\beta ,\beta ),(\alpha ,\beta \left)\right\}.$$

Then, $(Q,\precsim )$ is a complete lattice. Moreover, let us define the quantale operation ⊕ as follows:

Then, $(Q,\precsim ,\oplus )$ is a value quantale. Furthermore, equip Q with the QVQMS δ given by $\delta (\varrho ,\varsigma )=\varsigma \dot{-}\varrho$. According to this metric, the distances between the points can be easily computed, and they are presented in the table below:

Now, let us recall the definition of probabilistic metric space (see [3]). Let $\Phi$ be the collection of distribution functions satisfying monotonicity and left-continuity; that is, $\psi :[0,\infty )\to [0,1]$,

$$\forall x\in [0,\infty ),\underset{y<x}{sup}\psi \left(y\right)=\psi \left(x\right)\left(\mathrm{left-continuity}\right).$$

A mapping $\divideontimes :[0,1]\times [0,1]\to [0,1]$ is regarded as a left-continuous t-norm whenever the properties below hold:

• The operation ⋇ possesses both associativity and commutativity;
• $\lambda \divideontimes 1=\lambda$, for all $\lambda \in [0,1]$;
• Whenever ${\lambda }_1\le {\lambda }_2,{\lambda }_3\le {\lambda }_4$, it holds that ${\lambda }_1\divideontimes {\lambda }_3\le {\lambda }_2\divideontimes {\lambda }_4$;
• ${\lambda }_1\divideontimes {\lambda }_2=sup\{u\divideontimes v:0<u<{\lambda }_1,0<v<{\lambda }_2\}(\mathrm{left}\mathrm{continuity})$.

• See [3]. As stated in [3], a probabilistic metric is a mapping $\Psi :S\times S\to \Phi$ that meets the requirements given below for all $\alpha ,\beta ,\gamma \in S$ and ${\lambda }_1,{\lambda }_2\in [0,1]$:

• ${\Psi }_{\alpha ,\alpha }={ϵ}_0$;
• ${\Psi }_{\alpha ,\beta }\ne {ϵ}_0$ if $\alpha \ne \beta$;
• ${\Psi }_{\alpha ,\beta }={\Psi }_{\beta ,\alpha }$;
• ${\Psi }_{\alpha ,\gamma }({\lambda }_1+{\lambda }_2)\ge {\Psi }_{\alpha ,\beta }\left({\lambda }_1\right)\divideontimes {\Psi }_{\beta ,\gamma }\left({\lambda }_2\right)$.

• Here, $\Psi (\alpha ,\beta )={\Psi }_{\alpha ,\beta }$ and ${ϵ}_0$ is the top element of $\Phi$ according to the pointwise order and is defined by

  $${ϵ}_0\left(\lambda \right)=\left\{\begin{array}{cc}0,\hfill & \lambda =0\hfill \\ 1,\hfill & \lambda \ne 0\hfill \end{array}\right.$$

  and $(S,\Psi ,\divideontimes )$ is referred to as a probabilistic metric space (PMS). A sequence ${\left({\varrho }_n\right)}_{n\in \mathbb{N}}$ in S strongly converges to $\varrho \in S$ if $\zeta >0$ is given. There exists $\tilde{n}\in \mathbb{N}$, beyond which,

  $${\Psi }_{{\varrho }_n,\varrho }\left(\zeta \right)>1-\zeta \mathrm{whenever}n\ge \tilde{n}$$

  holds. It is termed a strong Cauchy when, for any $\zeta >0$, some $\tilde{n}\in \mathbb{N}$ can be found with

  $${\Psi }_{{\varrho }_n,{\varrho }_m}\left(\zeta \right)>1-\zeta \mathrm{whenever}m,n\ge \tilde{n}.$$

  Strong completeness of $(S,\Psi ,\divideontimes )$ means that any strong Cauchy sequence in S possesses a strong limit within S, as described in [3].

As shown in [4], PMSs can be obtained as a particular subclass within the framework of QVQMSs. In particular, consider the PMS $(S,\Psi ,\divideontimes )$, where ⋇ is a left-continuous t-norm. If we equip $\Phi$, the opposite pointwise order ${\le }^{op}$, defined by

$${\psi }_1{\le }^{op}{\psi }_2\iff {\psi }_1\left(x\right)\ge {\psi }_2\left(x\right),\mathrm{for}\mathrm{all}x\in [0,\infty )$$

and take the binary operation ⊕ given by

$$({\psi }_1\oplus {\psi }_2)\left(x\right)=\underset{y+z=x}{\bigvee }{\psi }_1\left(y\right)\divideontimes {\psi }_2\left(z\right)$$

(1)

as the quantale operation, then it follows that $(\Phi ,{\le }^{op},\oplus )$ is indeed a value quantale and $(S,\Psi )$ is a QVQMS. Moreover, a strong complete PMSs corresponds to an s-complete QVQMSs.

Definition 4 

([6]). Consider a value quantale $(Q,\precsim ,\oplus )$. An action of $[0,\infty ]$ on Q is a monotone mapping

$$\otimes :[0,\infty ]\times Q\to Q$$

such that, for all $\lambda ,\gamma \in [0,\infty ]$ and for all ${\varrho }_1,{\varrho }_2\in Q$, the following conditions hold:

1. 

$1\otimes {\varrho }_1={\varrho }_1$;

2. 

$\left(\lambda \gamma \right)\otimes {\varrho }_1=\lambda \otimes (\gamma \otimes {\varrho }_1)$;

3. 

$(\lambda +\gamma )\otimes {\varrho }_1=(\lambda \otimes {\varrho }_1)\oplus (\gamma \otimes {\varrho }_1)$.

Definition 5. 

Consider a value quantale $(Q,\precsim ,\oplus )$. A modified action of $[0,\infty ]$ on Q is a monotone mapping

$$\otimes :[0,\infty ]\times Q\to Q$$

such that, for all $\lambda ,\gamma \in [0,\infty ]$ and for all ${\varrho }_1,{\varrho }_2\in Q$, the following conditions hold:

1. 

$1\otimes {\varrho }_1={\varrho }_1$ and ${\varrho }_1\succsim \lambda \otimes {\varrho }_1$ for $\lambda \ne 1$;

2. 

$\left(\lambda \gamma \right)\otimes {\varrho }_1=\lambda \otimes (\gamma \otimes {\varrho }_1)$;

3. 

$(\lambda +\gamma )\otimes {\varrho }_1=(\lambda \otimes {\varrho }_1)\oplus (\gamma \otimes {\varrho }_1)$;

4. 

$(\lambda \otimes {\varrho }_1)\oplus (\lambda \otimes {\varrho }_2)\succsim \lambda \otimes ({\varrho }_1\oplus {\varrho }_2)$.

Remark 1. 

Throughout this paper, all computations are carried out with respect to the modified action.

The following examples serve as examples of both the action and the modified action.

Example 6 

([6]). Given $\lambda \in (0,1]$ and $\psi \in \Phi$, set

$$(\lambda \otimes \psi )\left(x\right)=\psi ({\displaystyle \frac{x}{\lambda }}),\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in [0,\infty ).$$

(2)

Then, ⊗ defines a modified action.

Example 7 

([7]). Let us consider the nonnegative extended real numbers $Q=[0,\infty ]$ with the usual ordering ≤ and usual addition + of $\mathbb{R}$. Then, $(Q,\le ,+)$ is a value quantale. The operation given below is a modified action of $(0,1)$ on Q:

$$\lambda \otimes a=\left\{\begin{array}{cc}\lambda a,\hfill & a<\infty \hfill \\ \infty ,\hfill & a=\infty ,\hfill \end{array}\right.$$

where $\lambda \in (0,1),a\in Q$.

Example 8. 

Consider the value quantale $(Q^{\prime },\le ,\oplus )$ given in Example 3. Define a mapping $\otimes :[0,1]\times Q^{\prime }\to Q^{\prime }$ by $(\lambda \otimes \zeta )\left(x\right)=\zeta \left(\lambda x\right)$, for all $\zeta \in Q^{\prime }$ and $\lambda \in [0,1]$. Then, ⊗ is a modified action of $[0,1]$ on $Q^{\prime }$. Indeed, for all $\lambda ,\gamma \in [0,1]$ and $\zeta ,\rho \in Q^{\prime }$, we have

1. 

$(1\otimes \zeta )\left(x\right)=\zeta \left(x\right)$, for all $x\in [0,\infty ]$. Then, $1\otimes \zeta =\zeta$. Let ${\lambda }^{\prime }\in [0,1)$. Then, ${\lambda }^{\prime }\otimes \zeta \le \zeta$, since ζ is nondecreasing.

2. 

$\left[\right(\lambda \gamma )\otimes \zeta ]\left(x\right)=\zeta \left(\lambda \gamma x\right)=[\lambda \otimes (\gamma \otimes \zeta \left)\right]\left(x\right)$ for all $x\in [0,\infty ]$, and hence

$$\left(\lambda \gamma \right)\otimes \zeta =\lambda \otimes (\gamma \otimes \zeta ).$$

3. 

$\begin{array}{cc}\left[\right(\lambda +\gamma )\otimes \zeta ]\left(x\right)=\zeta \left(\right(\lambda +\gamma \left)x\right)=\zeta (\lambda x+\gamma x)& =\lambda \zeta \left(x\right)+\gamma \zeta \left(x\right)\hfill \\ & =\left[\right(\lambda \otimes \zeta )\oplus (\gamma \otimes \zeta \left)\right]\left(x\right),\hfill \end{array}$for all $x\in [0,\infty ]$ and $(\lambda +\gamma )\otimes \zeta =(\lambda \otimes \zeta )\oplus (\gamma \otimes \zeta )$.

4. 

$[\lambda \otimes (\zeta \oplus \rho \left)\right]\left(x\right)=(\zeta \oplus \rho )\left(\lambda x\right)=\zeta \left(\lambda x\right)+\rho \left(\lambda x\right)=\left[\right(\lambda \otimes \zeta )\oplus (\lambda \otimes \rho \left)\right]\left(x\right)$, for all $x\in [0,\infty ]$. Consequently, $\lambda \otimes (\zeta \oplus \rho )=(\lambda \otimes \zeta )\oplus (\lambda \otimes \rho )$.

We conclude this section with the following example:

Example 9. 

Let $(Q,\precsim ,\oplus )$ be a value quantale, and let S be a nonempty set. Consider a quantale-valued quasi-metric space $(S,\delta )$. Let ⊗ be a modified action of $[0,\infty ]$ on Q, satisfying the following condition $\left(A\right)$ for all $\lambda \in (0,\infty ]$, and for all $\varrho ,\varsigma \in S$:

$$\left(A\right)\lambda \otimes \varrho =\lambda \otimes \varsigma \Rightarrow \varrho =\varsigma .$$

Fix $\lambda \in (1,\infty ]$. Define a mapping $\tilde{\delta }:S\times S\to Q$ by

$$\tilde{\delta }(\varrho ,\varsigma )=\lambda \otimes \delta (\varrho ,\varsigma )$$

for all $\varrho ,\varsigma \in S$. Then $(S,\tilde{\delta })$ is also a quantale-valued quasi-metric space. Indeed, for any $\varrho ,\varsigma ,\xi \in S$, the following hold:

• $\tilde{\delta }(\varrho ,\varrho )=\lambda \otimes \delta (\varrho ,\varrho )=\lambda \otimes \theta =\theta$, which follows from property 1 of the modified action ⊗.
• Let $\tilde{\delta }(\varrho ,\xi )=\tilde{\delta }(\xi ,\varrho )=\theta$. Then, from the property $\left(A\right)$ and the definition of $\tilde{\delta }$, we obtain $\delta (\varrho ,\xi )=\delta (\xi ,\varrho )=\theta$. Since δ is a quantale-valued quasi-metric, we obtain that $\varrho =\xi$.
• Finally, we establish the transitive property of $\tilde{\delta }$ using property 4 and the monotonicity of the modified action, as follows:

  $$\begin{array}{cc}\hfill \tilde{\delta }(\varrho ,\xi )& =\lambda \otimes \delta (\varrho ,\xi )\hfill \\ \hfill & \precsim \lambda \otimes \left[\delta \right(\varrho ,\varsigma )\oplus \delta (\varsigma ,\xi \left)\right]\hfill \\ \hfill & \precsim [\lambda \otimes \delta (\varrho ,\varsigma \left)\right]\oplus [\lambda \otimes \delta (\varsigma ,\xi \left)\right]\hfill \\ \hfill & =\tilde{\delta }(\varrho ,\varsigma )\oplus \tilde{\delta }(\varsigma ,\xi ).\hfill \end{array}$$

3. Results on Coupled Fixed Points Within QVQMS

For convenience, we shall denote the structure $(Q,\precsim ,\oplus )$ endowed with the modified action ⊗ by $(Q,\precsim ,\oplus ,\otimes )$ and we shall refer to $(Q,\precsim ,\oplus ,\otimes )$ as a quantale action structure in the paper.

As introduced in [16], the coupled fixed point has the following QVQMS-version:

Definition 6. 

Consider a value quantale $(Q,\precsim ,\oplus )$. Let $(S,\delta )$ be a QVQMS. The pair $(\varrho ,\varsigma )$ is a coupled fixed point of $P:S\times S\to S$ whenever

$$P(\varrho ,\varsigma )=\varrho ,P(\varsigma ,\varrho )=\varsigma .$$

Remark 2. 

The requirement (4) in Theorem 1 is motivated by the classical behaviour of geometric sequences in the real-valued metric setting. In the usual metric space $(\mathbb{R},\left|·\right|)$, the assumption $\lambda \in [0,1)$ implies that ${\lambda }^n\to 0$ as $n\to \infty$, which plays a fundamental role in contraction type fixed point theorems. In the quantale-valued metric framework, the structure is a complete lattice rather than a numerical space, and hence ordinary limits cannot be employed. For this reason, the convergence is expressed via the superior limit in complete lattices and this condition can be regarded as a quantale theoretic analogue of the classical property ${\lambda }^n\to 0$.

Theorem 1. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\delta )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\precsim \lambda \otimes \left[\delta \right(\varrho ,\xi )\vee \delta (\varsigma ,\tau \left)\right],$$

(3)

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

(4)

Then, P admits a unique coupled fixed point in S.

Proof. 

Let $({\varrho }_0,{\varsigma }_0)$ be a point satisfying condition (4) and set

$$\begin{array}{c}{\varrho }_1=P({\varrho }_0,{\varsigma }_0),{\varrho }_2=P({\varrho }_1,{\varsigma }_1),\dots ,{\varrho }_{n+1}=P({\varrho }_n,{\varsigma }_n)\\ \mathrm{and}\\ {\varsigma }_1=P({\varsigma }_0,{\varrho }_0),{\varsigma }_2=P({\varsigma }_1,{\varrho }_1),\dots ,{\varsigma }_{n+1}=P({\varsigma }_n,{\varrho }_n).\end{array}$$

Then, by condition (3), it follows that

$$\begin{array}{cc}\hfill \delta ({\varrho }_n,{\varrho }_{n+1})& =\delta (P({\varrho }_{n-1},{\varsigma }_{n-1}),P({\varrho }_n,{\varsigma }_n))\hfill \\ \hfill & \precsim \lambda \otimes [\delta ({\varrho }_{n-1},{\varrho }_n)\vee \delta ({\varsigma }_{n-1},{\varsigma }_n)].\hfill \end{array}$$

Analogously, we have

$$\begin{array}{cc}\hfill \delta ({\varsigma }_n,{\varsigma }_{n+1})& =\delta (P({\varsigma }_{n-1},{\varrho }_{n-1}),P({\varsigma }_n,{\varrho }_n))\hfill \\ \hfill & \precsim \lambda \otimes [\delta ({\varsigma }_{n-1},{\varsigma }_n)\vee \delta ({\varrho }_{n-1},{\varrho }_n)].\hfill \end{array}$$

Define ${\zeta }_n:=\delta ({\varrho }_n,{\varrho }_{n+1})\vee \delta ({\varsigma }_n,{\varsigma }_{n+1})$. Using the above inequalities and the monotonicity of the action ⊗, we have

$$\begin{array}{cc}\hfill {\zeta }_n& =\delta ({\varrho }_n,{\varrho }_{n+1})\vee \delta ({\varsigma }_n,{\varsigma }_{n+1})\hfill \\ \hfill & \precsim \lambda \otimes \left[\delta ({\varrho }_{n-1},{\varrho }_n)\vee \delta ({\varsigma }_{n-1},{\varsigma }_n)\right]\hfill \\ \hfill & =\lambda \otimes {\zeta }_{n-1}\hfill \\ & ⋮\hfill \\ \hfill & \precsim {\lambda }^n\otimes {\zeta }_0.\hfill \end{array}$$

When ${\zeta }_0=\theta$, one can readily see that $({\varrho }_0,{\varsigma }_0)$ is a coupled fixed point. Therefore, let us assume that ${\zeta }_0\gg \theta$. By the definition of the well-above relation and inequality (4), for any $ϵ\gg \theta$ one can find $n_0\in \mathbb{N}$ such that, whenever $n_0\le m<n$, we obtain

$$\begin{array}{c}\delta ({\varrho }_m,{\varrho }_n)\precsim \delta ({\varrho }_m,{\varrho }_{m+1})\oplus \dots \oplus \delta ({\varrho }_{n-1},{\varrho }_n)\\ \mathrm{and}\\ \delta ({\varsigma }_m,{\varsigma }_n)\precsim \delta ({\varsigma }_m,{\varsigma }_{m+1})\oplus \dots \oplus \delta ({\varsigma }_{n-1},{\varsigma }_n).\end{array}$$

Applying the supremum to each side of the above two inequalities and from property 3 of the modified action, we deduce

$$\begin{array}{cc}\hfill \delta ({\varrho }_m,{\varrho }_n)\vee \delta ({\varsigma }_m,{\varsigma }_n)& \precsim (\delta ({\varrho }_m,{\varrho }_{m+1})\vee \delta ({\varsigma }_m,{\varsigma }_{m+1}))\oplus \dots \oplus (\delta ({\varrho }_{n-1},{\varrho }_n)\vee \delta ({\varsigma }_{n-1},{\varsigma }_n))\hfill \\ \hfill & \precsim ({\lambda }^m\otimes {\zeta }_0)\oplus \dots \oplus ({\lambda }^{n-1}\otimes {\zeta }_0)\hfill \\ \hfill & \precsim ({\lambda }^m+\dots +{\lambda }^{n-1})\otimes {\zeta }_0\hfill \\ \hfill & \precsim {\displaystyle \frac{{\lambda }^m}{1-\lambda }}\otimes {\zeta }_0\precsim {\displaystyle \frac{{\lambda }^m}{1-\lambda }}\otimes [{\delta }^s({\varrho }_0,{\varrho }_1)\vee {\delta }^s({\varsigma }_0,{\varsigma }_1)]\precsim ϵ.\hfill \end{array}$$

Thus, we have ${\delta }^s({\varrho }_m,{\varrho }_n)\precsim ϵ$ and ${\delta }^s({\varsigma }_m,{\varsigma }_n)\precsim ϵ$. Hence, the sequences $\left({\varrho }_n\right)$ and $\left({\varsigma }_n\right)$ are Cauchy. Since $(S,\delta )$ is s-complete, there exist points $\varrho *,\varsigma *\in S$, such that ${\varrho }_n\to \varrho *$ and ${\varsigma }_n\to \varsigma *$, with respect to ${\delta }^s$. According to Theorem 2.9 in [4], for each $ϵ\gg \theta$ one can find $\mu \gg \theta$ with $2\mu =\mu \oplus \mu \ll ϵ$ and $n_0\in \mathbb{N}$, such that, for all $n\ge n_0$, ${\delta }^s({\varrho }_n,\varrho *)\precsim \mu$ and ${\delta }^s({\varsigma }_n,\varsigma *)\precsim \mu$. We now show that $(\varrho *,\varsigma *)$ is a coupled fixed point of P:

$$\begin{array}{cc}\hfill {\delta }^s(P(\varrho *,\varsigma *),\varrho *)& \precsim {\delta }^s(P(\varrho *,\varsigma *),{\varrho }_{n+1})\oplus {\delta }^s({\varrho }_{n+1},\varrho *)\hfill \\ \hfill & \precsim {\delta }^s(P(\varrho *,\varsigma *),P({\varrho }_n,{\varsigma }_n))\oplus {\delta }^s({\varrho }_{n+1},\varrho *)\hfill \\ \hfill & \precsim [\lambda \otimes ({\delta }^s(\varrho *,{\varrho }_n)\vee {\delta }^s(\varsigma *,{\varsigma }_n))]\oplus {\delta }^s({\varrho }_{n+1},\varrho *)\hfill \\ \hfill & \precsim (\lambda \otimes \mu )\oplus \mu \hfill \\ \hfill & \precsim \mu \oplus \mu \precsim ϵ.\hfill \end{array}$$

Thus, we obtain $P(\varrho *,\varsigma *)=\varrho *$. Moreover, in a similar way, $P(\varsigma *,\varrho *)=\varsigma *$. What is left is to prove that the uniqueness. Suppose, to the contrary, that $({\varrho }^{\prime },{\varsigma }^{\prime })$ is a distinct coupled fixed point of P. Then, we have

$$\begin{array}{cc}\hfill \delta ({\varrho }^{\prime },\varrho *)& =\delta (P({\varrho }^{\prime },{\varsigma }^{\prime }),P(\varrho *,\varsigma *))\hfill \\ \hfill & \precsim \lambda \otimes [\delta ({\varrho }^{\prime },\varrho *)\vee \delta ({\varsigma }^{\prime },\varsigma *)]\hfill \\ & \mathrm{and}\\ \hfill \delta ({\varsigma }^{\prime },\varsigma *)& =\delta (P({\varsigma }^{\prime },{\varrho }^{\prime }),P(\varsigma *,\varrho *))\hfill \\ \hfill & \precsim \lambda \otimes [\delta ({\varsigma }^{\prime },\varsigma *)\vee \delta ({\varrho }^{\prime },\varrho *)].\hfill \end{array}$$

From the above inequalities, we obtain

$$\delta ({\varrho }^{\prime },\varrho *)\vee \delta ({\varsigma }^{\prime },\varsigma *)\precsim \lambda \otimes [\delta ({\varsigma }^{\prime },\varsigma *)\vee \delta ({\varrho }^{\prime },\varrho *)].$$

Since, $1\otimes [\delta ({\varrho }^{\prime },\varrho *)\vee \delta ({\varsigma }^{\prime },\varsigma *)]=\delta ({\varrho }^{\prime },\varrho *)\vee \delta ({\varsigma }^{\prime },\varsigma *)$ and the action ⊗ is monotone, the above inequality yields a contradiction. Consequently, P has exactly one coupled fixed point. □

Corollary 1. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\rho )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\precsim (\alpha \otimes \delta (\varrho ,\xi \left)\right)\oplus (\beta \otimes \delta (\varsigma ,\tau \left)\right),$$

(5)

where $\alpha ,\beta \ge 0$ and $\lambda =\alpha +\beta <1$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

(6)

Then, P admits a unique coupled fixed point in S.

Proof. 

We only need to verify that the contraction assumption (5) yields the one in (3). Let $\alpha ,\beta \ge 0$ be constants satisfying $\alpha +\beta <1$. Take arbitrary elements $\varrho ,\varsigma ,\xi ,\tau \in S$, satisfying (5). Then, we have

$$\begin{array}{cc}\hfill \delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)& \precsim (\alpha \otimes \delta (\varrho ,\xi \left)\right)\oplus (\beta \otimes \delta (\varsigma ,\tau \left)\right)\hfill \\ \hfill & \precsim [\alpha \otimes (\delta (\varrho ,\xi )\vee \delta (\varsigma ,\tau )\left)\right]\oplus [\beta \otimes (\delta (\varrho ,\xi )\vee \delta (\varsigma ,\tau )\left)\right]\hfill \\ \hfill & \precsim (\alpha +\beta )\otimes \left[\delta \right(\varrho ,\xi )\vee \delta (\varsigma ,\tau \left)\right].\hfill \end{array}$$

Hence, the proof follows from Theorem 1. □

Corollary 2. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\delta )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta (P(\varrho ,\varsigma ),P(\xi ,\tau ))\precsim {\displaystyle \frac{\lambda }{2}}\otimes [\delta (\varrho ,\xi )\oplus \delta (\varsigma ,\tau )],$$

(7)

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

(8)

Then, P admits a unique coupled fixed point in S.

Proof. 

Take arbitrary $\varrho ,\varsigma ,\xi ,\tau \in S$ satisfying (7). Let $\lambda \in [0,1)$. Then, we obtain the following:

$$\begin{array}{cc}\hfill \delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)& \precsim {\displaystyle \frac{\lambda }{2}}\otimes [\delta (\varrho ,\xi )\oplus \delta (\varsigma ,\tau )]\hfill \\ \hfill & \precsim [{\displaystyle \frac{\lambda }{2}}\otimes \delta (\varrho ,\xi )]\oplus [{\displaystyle \frac{\lambda }{2}}\otimes \delta (\varsigma ,\tau )]\hfill \\ \hfill & \precsim [{\displaystyle \frac{\lambda }{2}}\otimes (\delta (\varrho ,\xi )\vee \delta (\varsigma ,\tau ))]\oplus [{\displaystyle \frac{\lambda }{2}}\otimes (\delta (\varsigma ,\tau )\vee \delta (\varrho ,\xi ))]\hfill \\ \hfill & =\lambda \otimes \left(\delta \right(\varrho ,\xi )\vee \delta (\varsigma ,\tau \left)\right).\hfill \end{array}$$

This completes the proof by Theorem 1. □

Remark 3. 

Let $Q=[0,\infty ]$ be equipped with the usual order on the extended non-negative real numbers and with the ordinary addition as the quantale operation. In this case, quantale-valued quasi-metric spaces reduce to classical extended quasi-metric spaces. If a quasi-metric space is complete with respect to its symmetrization, then it is called a bicomplete quasi-metric space (see [26]). Hence, every bicomplete extended quasi-metric space can be viewed as an s-complete quantale-valued quasi-metric space. Moreover, consider the action ⊗ of $[0,1)$ on $[0,\infty ]$ given by

$$\lambda \otimes r=\left\{\begin{array}{cc}\lambda r,\hfill & r<\infty \hfill \\ \infty ,\hfill & r=\infty ,\hfill \end{array}\right.$$

for $\lambda \in [0,1)$ and $r\in [0,\infty ]$ (see [7]). Then, the following corollaries follow directly from the main results.

Corollary 3. 

Consider an extended bicomplete quasi-metric space $(S,\delta )$. Assume that the mapping $P:S\times S\to S$ satisfies the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\le \lambda max\left\{\delta \right(\varrho ,\xi ),\delta (\varsigma ,\tau \left)\right\},$$

(9)

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$${\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))<\infty .$$

(10)

Then, P admits a unique coupled fixed point in S.

Proof. 

Let $(S,\delta )$ be an extended bicomplete quasi-metric space, where $\delta :S\times S\to [0,\infty ]$. Then, from Remark 3, $\left(\right[0,\infty ],\le ,+,\otimes )$ is a quantale action structure and $(S,\delta )$ is an s-complete quantale-valued quasi-metric space. By the definition of the action ⊗, condition (9) yields condition (3). Now, consider a point $({\varrho }_0,{\varsigma }_0)$ satisfying condition (10). Then,

$$\underset{n\to \infty }{lim}{\lambda }^n({\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0)))=0.$$

Moreover,

$$\underset{n\to \infty }{lim\; sup}{\lambda }^n({\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0)))=0,$$

that is,

$$\underset{n\to \infty }{lim\; sup}{\lambda }^n\otimes ({\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0)))=0.$$

Therefore, condition (10) implies condition (4), and by Theorem 1, P admits a unique coupled fixed point in S. □

Corollary 4. 

Consider an extended bicomplete quasi-metric space $(S,\delta )$. Assume that the mapping $P:S\times S\to S$ satisfies the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\le \alpha \delta (\varrho ,\xi )+\beta \delta (\varsigma ,\tau ),$$

where $\alpha ,\beta \ge 0$ and $\alpha +\beta <1$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$${\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))<\infty .$$

Then, P admits a unique coupled fixed point in S.

Proof. 

By following the steps of the proof of Corollary 3, this result follows as a consequence of Corollary 1; hence, the proof is omitted. □

In [18], Berinde obtained a generalization of the results of Bhaskar and Lakshmikantham [16]. Motivated by Theorem 3 in [18], we now present, below, a generalized version of Theorem 1.

Theorem 2. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\delta )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\vee \delta \left(P\right(\varsigma ,\varrho ),P(\tau ,\xi \left)\right)\precsim \lambda \otimes \left[\delta \right(\varrho ,\xi )\vee \delta (\varsigma ,\tau \left)\right],$$

(11)

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

(12)

Then, P admits a unique coupled fixed point in S.

Proof. 

Let $({\varrho }_0,{\varsigma }_0)$ be a point fulfilling the condition (12), and let $\delta *:S^2\times S^2\to Q$ be defined by

$${\delta }_{\times }((\varrho ,\varsigma ),(\xi ,\tau ))=\delta (\varrho ,\xi )\vee \delta (\varsigma ,\tau )$$

for all $(\varrho ,\varsigma ),(\xi ,\tau )\in S^2$. Then, ${\delta }_{\times }$ is a quantale-valued quasi-metric on $S^2$ (see [25]) and $(S^2,{\delta }_{\times })$ is an s-complete QVQMS. We now define the mapping $P*:S^2\to S^2$ as follows:

$$P*(\varrho ,\varsigma )=(P(\varrho ,\varsigma ),P(\varsigma ,\varrho ))$$

for all $(\varrho ,\varsigma )\in S^2$. Explicitly, for all $(\varrho ,\varsigma ),(\xi ,\tau )\in S^2$, one has

$${\delta }_{\times }(P^{*}(\varrho ,\varsigma ),P^{*}(\xi ,\tau ))=\delta (P(\varrho ,\varsigma ),P(\xi ,\tau ))\vee \delta (P(\varsigma ,\varrho ),P(\tau ,\xi ))$$

and

$${\delta }_{\times }((\varrho ,\varsigma ),(\xi ,\tau ))=\delta (\varrho ,\xi )\vee \delta (\varsigma ,\tau ).$$

Then, condition (11) yields the following:

$${\delta }_{\times }(P^{*}(\varrho ,\varsigma ),P^{*}(\xi ,\tau ))\precsim \lambda \otimes {\delta }_{\times }((\varrho ,\varsigma ),(\xi ,\tau )),$$

for all $(\varrho ,\varsigma ),(\xi ,\tau )\in S^2$, where $\lambda \in [0,1)$. Let ${\alpha }_0=({\varrho }_0,{\varsigma }_0)$, and consider the iteration sequence $\left({\alpha }_n\right)\subset S^2$ defined by

$${\alpha }_{n+1}=P*\left({\alpha }_n\right),n\ge 0$$

with ${\alpha }_n=({\varrho }_n,{\varsigma }_n)$. Then, we have

$${\delta }_{\times }(P^{*}\left({\alpha }_n\right),P^{*}\left({\alpha }_{n-1}\right))\precsim \lambda \otimes {\delta }_{\times }({\alpha }_n,{\alpha }_{n-1}),n\ge 1.$$

By induction, together with the property 2 and the monotonicity of the action ⊗, we obtain

$$\begin{array}{cc}\hfill {\delta }_{\times }(P^{*}\left({\alpha }_n\right),P^{*}\left({\alpha }_{n-1}\right))& \precsim \lambda \otimes {\delta }_{\times }({\alpha }_n,{\alpha }_{n-1})\hfill \\ \hfill & \precsim \lambda \otimes (\lambda \otimes {\delta }_{\times }({\alpha }_{n-1},{\alpha }_{n-2}))\hfill \\ \hfill & ={\lambda }^2\otimes {\delta }_{\times }({\alpha }_{n-1},{\alpha }_{n-2})\hfill \\ & ⋮\hfill \\ \hfill & \precsim {\lambda }^n\otimes {\delta }_{\times }({\alpha }_0,{\alpha }_1).\hfill \end{array}$$

Moreover, one has that ${\delta }_{\times }^s(P^{*}\left({\alpha }_n\right),P^{*}\left({\alpha }_{n-1}\right))\precsim {\lambda }^n\otimes {\delta }_{\times }^s({\alpha }_0,{\alpha }_1)$. We now prove that $\left({\alpha }_n\right)$ is a Cauchy sequence in $(S^2,{\delta }_{\times })$. By the well above relation and from the condition (12), for any $ϵ\gg \theta$, we can find $n_0\in \mathbb{N}$, such that for all $\tilde{n}\ge n_0$,

$$ϵ\succsim {\lambda }^{\tilde{n}}\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))].$$

Fix $m,n\in \mathbb{N}$ with $n_0<m<n$. Then, by properties 1 and 3 of the modified action ⊗, we obtain that

$$\begin{array}{cc}\hfill {\delta }_{\times }^s({\alpha }_m,{\alpha }_n)& \precsim {\delta }_{\times }^s({\alpha }_m,{\alpha }_{m+1})\oplus \dots \oplus {\delta }_{\times }^s({\alpha }_{n-1},{\alpha }_n)\hfill \\ \hfill & \precsim ({\lambda }^m\otimes {\delta }_{\times }^s({\alpha }_0,{\alpha }_1))\oplus \dots \oplus ({\lambda }^{n-1}\otimes {\delta }_{\times }^s({\alpha }_0,{\alpha }_1))\hfill \\ \hfill & \precsim ({\lambda }^m+\dots +{\lambda }^{n-1})\otimes {\delta }_{\times }^s({\alpha }_0,{\alpha }_1)\hfill \\ \hfill & \precsim {\displaystyle \frac{{\lambda }^m}{1-\lambda }}\otimes {\delta }_{\times }^s({\alpha }_0,{\alpha }_1)={\displaystyle \frac{{\lambda }^m}{1-\lambda }}\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]\hfill \\ \hfill & \precsim {\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]\hfill \\ \hfill & \precsim ϵ.\hfill \end{array}$$

Thus, the sequence $\left({\alpha }_n\right)$ is Cauchy. Since $(S^2,{\delta }_{\times })$ is s-complete, there exists point $\alpha =({\varrho }^{*},{\varsigma }^{*})\in S^2$, such that ${\alpha }_n\to \alpha$ with respect to ${\delta }_{\times }^s$. As stated in the proof of Theorem 1, for each $ϵ\gg \theta$, one can find $\mu \gg \theta$ with $2\mu =\mu \oplus \mu \ll ϵ$ and $n_0\in \mathbb{N}$ such that, for all $n\ge n_0$, ${\delta }_{\times }^s({\alpha }_n,\alpha )\precsim \mu$. We now show that $\alpha =(\varrho *,\varsigma *)$ is a fixed point of $P^{*}$:

$$\begin{array}{cc}\hfill {\delta }_{\times }^s(P^{*}\left(\alpha \right),\alpha )& \precsim {\delta }_{\times }^s(P^{*}\left(\alpha \right),{\alpha }_{n+1})\oplus {\delta }_{\times }^s({\alpha }_{n+1},\alpha )\hfill \\ \hfill & ={\delta }_{\times }^s(P^{*}\left(\alpha \right),P{(}^{*}{\alpha }_n))\oplus {\delta }_{\times }^s({\alpha }_{n+1},\alpha )\hfill \\ \hfill & \precsim (\lambda \otimes {\delta }_{\times }^s(\alpha ,{\alpha }_n))\oplus {\delta }_{\times }^s({\alpha }_{n+1},\alpha )\hfill \\ \hfill & \precsim {\delta }_{\times }^s(\alpha ,{\alpha }_n)\oplus {\delta }_{\times }^s({\alpha }_{n+1},\alpha )\hfill \\ \hfill & \precsim \mu \oplus \mu \hfill \\ \hfill & \precsim ϵ.\hfill \end{array}$$

Since $ϵ\gg \theta$ is arbitrary, we see that ${\delta }_{\times }^s(P^{*}\left(\alpha \right),\alpha )=\theta$ and $\alpha$ is a unique fixed point of $P^{*}$. Thus, $(\varrho *,\varsigma *)$ is a unique coupled fixed point of P. □

As a consequence of Theorem 2, we can state the following corollaries. Since their proofs are very similar to those of Corollary 1 and Corollary 2, respectively, they are omitted.

Corollary 5. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\rho )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta \left(P\right(\varrho ,\varsigma ),P(\xi ,\tau \left)\right)\vee \delta \left(P\right(\varsigma ,\varrho ),P(\tau ,\xi \left)\right)\precsim (\alpha \otimes \delta (\varrho ,\xi \left)\right)\oplus (\beta \otimes \delta (\varsigma ,\tau \left)\right),$$

where $\alpha ,\beta \ge 0$ and $\lambda =\alpha +\beta <1$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

Then, P admits a unique coupled fixed point in S.

Corollary 6. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,\delta )$ be an s-complete QVQMS. Consider a mapping $P:S\times S\to S$ satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$\delta (P(\varrho ,\varsigma ),P(\xi ,\tau ))\vee \delta (P(\varsigma ,\varrho ),P(\tau ,\xi ))\precsim {\displaystyle \frac{\lambda }{2}}\otimes [\delta (\varrho ,\xi )\oplus \delta (\varsigma ,\tau )],$$

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

Then, P admits a unique coupled fixed point in S.

Furthermore, the following results for bicomplete extended quasi-metric spaces follow as consequences of Corollaries 5 and 6.

Corollary 7. 

Consider an extended bicomplete quasi-metric space $(S,\delta )$. Assume that the mapping $P:S\times S\to S$ satisfies the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$max\left\{\delta \right(P(\varrho ,\varsigma ),P(\xi ,\tau )),\delta (P(\varsigma ,\varrho ),P(\tau ,\xi )\left)\right\}\le \alpha \delta (\varrho ,\xi )+\beta \delta (\varsigma ,\tau ),$$

where $\alpha ,\beta \ge 0$ and $\alpha +\beta <1$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$${\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))<\infty .$$

Then, P admits a unique coupled fixed point in S.

Corollary 8. 

Consider an extended bicomplete quasi-metric space $(S,\delta )$. Assume that the mapping $P:S\times S\to S$ satisfies the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$$max\{\delta (P(\varrho ,\varsigma ),P(\xi ,\tau )),\delta (P(\varsigma ,\varrho ),P(\tau ,\xi ))\}\le {\displaystyle \frac{\lambda }{2}}[\delta (\varrho ,\xi )+\delta (\varsigma ,\tau )],$$

where $\lambda \in [0,1)$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$${\delta }^s({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }^s({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))<\infty .$$

Then, P admits a unique coupled fixed point in S.

4. Applications Within QVPMS and PMS

QVPMs were defined in [23] as follows:

Definition 7 

(Kopperman et al., see [23]). Consider a nonempty set S and value quantale $(Q,\precsim ,\oplus )$. A mapping ${\delta }_p:S\times S\to Q$ is referred as a quantale-valued partial metric whenever, for all $\varrho ,\upsilon ,\varsigma \in S$, the conditions below are satisfied:

• $\left(Q{P}_1\right)$${\delta }_p(\varrho ,\varsigma )\succsim {\delta }_p(\varrho ,\varrho )$;
• $\left(Q{P}_2\right)$${\delta }_p(\varrho ,\varsigma )={\delta }_p(\varsigma ,\varrho )$;
• $\left(Q{P}_3\right)$$\varrho =\varsigma \mathit{iff}{\delta }_p(\varrho ,\varsigma )={\delta }_p(\varrho ,\varrho )={\delta }_p(\varsigma ,\varsigma )$;
• $\left(Q{P}_4\right)$${\delta }_p(\varrho ,\upsilon )\precsim {\delta }_p(\varrho ,\varsigma )\oplus \left[{\delta }_p(\varsigma ,\upsilon )\dot{-}{\delta }_p(\varsigma ,\varsigma )\right]$.
• The pair $(S,{\delta }_p)$ is referred as a quantale-valued partial metric space (QVPMS).

The notion of an 0-complete partial metric, originally given by Romaguera in [22], will be carried over to the framework QVPMSs as follows:

Definition 8. 

Consider a nonempty set S and value quantale $(Q,\precsim ,\oplus )$. Let $(S,{\delta }_p)$ be a QVPMS. A sequence $\left({\varrho }_n\right)\in S$ is called θ-Cauchy if

$$\underset{n,m\to \infty }{lim}{\delta }_p({\varrho }_n,{\varrho }_m)=\theta ,$$

that is, for any $ϵ\gg \theta$, there exists $n_0\in \mathbb{N}$, such that for all $m,n\ge n_0$, ${\delta }_p({\varrho }_m,{\varrho }_n)\precsim ϵ$. If every θ-Cauchy sequence $\left({\varrho }_n\right)$ converges to a point $\varrho *\in S$, such that ${\delta }_p(\varrho *,\varrho *)=\theta$, i.e., for any $ϵ\gg \theta$, there exists $n_0\in \mathbb{N}$, such that for all $n\ge n_0$, ${\delta }_p({\varrho }_n,\varrho *)\precsim ϵ$, then $(S,{\delta }_p)$ is called ${\theta }_s$-complete QVPMS.

Example 10. 

Consider a value quantale $(Q,\precsim ,\oplus )$. Let us define a mapping ${\delta }_p:Q\times Q\to Q$ by ${\delta }_p(\varrho ,\varsigma )=\varrho \vee \varsigma$ for all $\varrho ,\varsigma \in S$. Then, $(Q,{\delta }_p)$ is ${\theta }_s$-complete QVPMS. Indeed, we first establish that $(Q,{\delta }_p)$ is a QVPMS. Let $\varrho ,\varsigma ,\nu \in Q$:

• $\left(Q{P}_1\right)$${\delta }_p(\varrho ,\varrho )=\varrho \precsim {\delta }_p(\varrho ,\varsigma )=\varrho \vee \varsigma$;
• $\left(Q{P}_2\right)$${\delta }_p(\varrho ,\varsigma )=\varrho \vee \varsigma =\varsigma \vee \varrho ={\delta }_p(\varsigma ,\varrho )$;
• $\left(Q{P}_3\right)$${\delta }_p(\varrho ,\varsigma )={\delta }_p(\varrho ,\varrho )={\delta }_p(\varsigma ,\varsigma )\iff \varrho =\varrho \vee \varsigma =\varsigma \iff \varrho =\varsigma$.
• $\left(Q{P}_4\right)$ It is clear that the following inequality can be derived from the definition of supremum and condition $\left(q_4\right)$ of Definition 2.1 in [4]:

  $$\begin{array}{c}\hfill \varrho \precsim (\varrho \vee \varsigma )\oplus \left[\right(\varsigma \vee \nu \left)\dot{-}\right(\varsigma \vee \varsigma \left)\right].\end{array}$$

  (13)

  Moreover, from the (2) property of Theorem 2.2 in [4], one can obtain:

  $$\begin{array}{cc}\hfill \nu \precsim \nu \vee \varsigma & \precsim (\varrho \vee \varsigma )\oplus \left[\right(\varsigma \vee \nu \left)\dot{-}\right(\varsigma \vee \varrho \left)\right]\hfill \\ \hfill & \precsim (\varrho \vee \varsigma )\oplus \left[\right(\varsigma \vee \nu \left)\dot{-}\right(\varsigma \vee \varsigma \left)\right].\hfill \end{array}$$

  (14)

  If the supremum of both sides of inequalities (7) and (8) is taken, the following is obtained:

  $$\begin{array}{c}\hfill \varrho \vee \nu \precsim (\varrho \vee \varsigma )\oplus \left[\right(\varsigma \vee \nu \left)\dot{-}\right(\varsigma \vee \varsigma \left)\right].\end{array}$$

  Therefore, we conclude that ${\delta }_p$ is a quantale-valued partial metric. Next, we prove that $(Q,{\delta }_p)$ is ${\theta }_s$-complete. Let $\left({\varrho }_n\right)$ be a θ-Cauchy sequence. Then, for any $ϵ\gg \theta$, there exists $n_0\in \mathbb{N}$, such that for all $n_0\le m,n$,

  $${\delta }_p({\varrho }_m,{\varrho }_n)={\varrho }_m\vee {\varrho }_n\precsim ϵ.$$

  Since $ϵ\gg \theta$ is arbitrary and from the completely distributive lattice property of Q, we have

  $$\theta =inf\{q\in Q:q\gg \theta \}.$$

  Thus, it follows that ${\varrho }_n,{\varrho }_m=\theta$ for all $m,n\ge n_0$. Consequently, $\left({\varrho }_n\right)$ converges to θ, and since ${\delta }_p(\theta ,\theta )=\theta$, we deduce that $(Q,{\delta }_p)$ is a ${\theta }_s$-complete QVPMS.

In the subsequent results, we draw inspiration from the work presented in [24]. Our first aim is to demonstrate that Proposition 2.1 in [24] can be generalized to the framework of QVPMSs.

Proposition 1. 

Consider a value quantale $(Q,\precsim ,\oplus )$. Suppose that $(S,{\delta }_p)$ is a ${\theta }_s$-complete QVPMS. Define a mapping ${\delta }^{*}:S\times S\to Q$ by

$${\delta }^{*}(\varrho ,\varsigma )=\left\{\begin{array}{cc}\theta ,\hfill & \varrho =\varsigma \hfill \\ {\delta }_p(\varrho ,\varsigma ),\hfill & \varrho \ne \varsigma \hfill \end{array}\right.$$

Then, $(S,{\delta }^{*})$ is an s-complete and symmetric QVQMS.

Proof. 

First, we show that $(S,\delta *)$ is a symmetric QVQMS. We only verify the triangle inequality, since the other conditions are straightforward. Let $\varrho ,\varsigma ,\nu \in S$. We now examine the following cases:

• Case I. Let $\varrho \ne \varsigma =\nu$. Then we have

  $$\begin{array}{cc}\hfill {\delta }^{*}(\varrho ,\varsigma )={\delta }_p(\varrho ,\varsigma )& \precsim {\delta }_p(\varrho ,\nu )\oplus \left[{\delta }_p(\nu ,\varsigma )\dot{-}{\delta }_p(\nu ,\nu )\right]\hfill \\ \hfill & \precsim {\delta }^{*}(\varrho ,\nu )\oplus \left[{\delta }_p(\nu ,\nu )\dot{-}{\delta }_p(\nu ,\nu )\right]\hfill \\ \hfill & \precsim {\delta }^{*}(\varrho ,\nu )\oplus \theta ={\delta }^{*}(\varrho ,\nu )\oplus {\delta }^{*}(\nu ,\varsigma ).\hfill \end{array}$$
• Case II. Let $\varrho =\nu \ne \varsigma$. It follows that

  $$\begin{array}{cc}\hfill {\delta }^{*}(\varrho ,\varsigma )={\delta }_p(\varrho ,\varsigma )& \precsim {\delta }_p(\varrho ,\nu )\oplus \left[{\delta }_p(\nu ,\varsigma )\dot{-}{\delta }_p(\nu ,\nu )\right]\hfill \\ \hfill & \precsim \theta \oplus {\delta }_p(\nu ,\varsigma )={\delta }^{*}(\varrho ,\nu )\oplus {\delta }^{*}(\nu ,\varsigma ).\hfill \end{array}$$
• Case III. Let $\varrho \ne \varsigma \ne \nu$. In this case, we obtain

  $$\begin{array}{cc}\hfill {\delta }^{*}(\varrho ,\varsigma )={\delta }_p(\varrho ,\varsigma )& \precsim {\delta }_p(\varrho ,\nu )\oplus \left[{\delta }_p(\nu ,\varsigma )\dot{-}{\delta }_p(\nu ,\nu )\right]\hfill \\ \hfill & \precsim {\delta }^{*}(\varrho ,\nu )\oplus \left[{\delta }^{*}(\nu ,\varsigma )\dot{-}{\delta }^{*}(\nu ,\nu )\right]\hfill \\ \hfill & \precsim {\delta }^{*}(\varrho ,\nu )\oplus {\delta }^{*}(\nu ,\varsigma ).\hfill \end{array}$$
• Case IV. Finally, we consider when $\varrho =\varsigma$, it easy to see that

  $$\begin{array}{c}\hfill {\delta }^{*}(\varrho ,\varsigma )=\theta \precsim {\delta }^{*}(\varrho ,\nu )\oplus {\delta }^{*}(\nu ,\varsigma ).\end{array}$$

We now prove that S is s-complete. Let $\left({\varrho }_n\right)$ be a Cauchy sequence in $(S,\delta *)$. We can distinguish two possible situations below:

Case I: Assume that only a finite number of terms of the sequence coincide; that is,

$${\varrho }_{n_1}={\varrho }_{n_2}=\dots ={\varrho }_{n_m}$$

for some indices $n_1,n_2,\dots ,n_m\in \mathbb{N}$. Let us define $n_0:=max\{n_1,n_2,\dots ,n_m\}$. Then, for all $m,n\ge n_0$, the terms of the sequence are distinct. Consequently, $\delta *({\varrho }_n,{\varrho }_m)={\delta }_p({\varrho }_n,{\varrho }_m)$ for all $m,n\ge n_0$ and $\left({\varrho }_n\right)$ is a $\theta$-Cauchy in $(S,{\delta }_p)$.

Case II: If an infinite number of terms of the sequence coincide, then the sequence is eventually constant and hence, it is convergent.

Combining both cases, it is sufficient to consider the case where ${\varrho }_n\ne {\varrho }_m$ holds whenever $n\ne m$. Then, $\delta *({\varrho }_n,{\varrho }_m)={\delta }_p({\varrho }_n,{\varrho }_m)$ and hence, $\left({\varrho }_n\right)$ is a $\theta$-Cauchy in $(S,{\delta }_p)$. Because S is ${\theta }_s$-complete, we can guarantee the existence of a point $\varrho *\in S$, such that the sequence converges to ${\varrho }^{*}$ with respect to ${\delta }_p$. Therefore, for any $ϵ\gg \theta$, there exists $n_0\in \mathbb{N}$, such that for all $m,n\ge n_0$, we have ${\delta }_p({\varrho }_n,{\varrho }^{*})\precsim ϵ$ and hence, $\delta *({\varrho }_n,{\varrho }^{*})\precsim ϵ,\delta *({\varrho }^{*},{\varrho }_n)\precsim ϵ$. Consequently, we find that $(S,\delta *)$ is an s-complete QVQMS. □

Corollary 9. 

Consider a quantale action structure $(Q,\precsim ,\oplus ,\otimes )$, and let $(S,{\delta }_p)$ be an ${\theta }_s$-complete QVPMS. Let $P:S\times S\to S$ be a mapping satisfying the following condition for all $\varrho ,\varsigma ,\xi ,\tau \in S$:

$${\delta }_p(P(\varrho ,\varsigma ),P(\xi ,\tau ))\precsim (\alpha \otimes {\delta }_p(\varrho ,\xi ))\oplus (\beta \otimes {\delta }_p(\varsigma ,\tau )),$$

(15)

where $\alpha ,\beta \ge 0\mathit{with}\lambda =\alpha +\beta <1$. Suppose that there exists a couple $({\varrho }_0,{\varsigma }_0)$ satisfying the following requirement:

$$\underset{n\in \mathbb{N}}{\bigwedge }\underset{m\ge n}{\bigvee }{\lambda }^m\otimes [{\delta }_p({\varrho }_0,P({\varrho }_0,{\varsigma }_0))\vee {\delta }_p({\varsigma }_0,P({\varsigma }_0,{\varrho }_0))]=\theta .$$

(16)

Then, P admits a unique coupled fixed point in S.

Proof. 

Proposition 1 ensures that $(S,\delta *)$ is an s-complete and symmetric QVQMS; accordingly, it suffices to show that contraction condition (15) implies condition (3), which completes the proof. Let $\varrho ,\varsigma ,\xi ,\tau \in S$ and suppose that (15) holds. Then, we have

$$\begin{array}{cc}\hfill {\delta }_p(P(\varrho ,\varsigma ),P(\xi ,\tau ))& \precsim (\alpha \otimes {\delta }_p(\varrho ,\xi ))\oplus (\beta \otimes {\delta }_p(\varsigma ,\tau ))\hfill \\ \hfill & \precsim \lambda \otimes [{\delta }_p(\varrho ,\xi )\vee {\delta }_p(\varsigma ,\tau )],\hfill \end{array}$$

where $\lambda =\alpha +\beta$. If $P(\varrho ,\varsigma )=P(\xi ,\tau )$, the proof is straightforward; hence, we may assume that $P(\varrho ,\varsigma )\ne P(\xi ,\tau )$. Therefore, we will examine the following conditions:

• Case I. Let $\varrho \ne \xi \ne$ and $\varsigma \ne \tau$. Then, we obtain

  $$\begin{array}{cc}\hfill {\delta }^{*}(P(\varrho ,\varsigma ),P(\xi ,\tau ))={\delta }_p(P(\varrho ,\varsigma ),P(\xi ,\tau ))& \precsim \lambda \otimes [{\delta }_p(\varrho ,\xi )\vee {\delta }_p(\varsigma ,\tau )]\hfill \\ \hfill & \precsim \lambda \otimes [{\delta }^{*}(\varrho ,\xi )\vee {\delta }^{*}(\varsigma ,\tau )].\hfill \end{array}$$
• Case II. Let $\varrho =\xi$ and $\varsigma \ne \tau$. This implies that

  $$\begin{array}{cc}\hfill {\delta }^{*}(P(\varrho ,\varsigma ),P(\xi ,\tau ))={\delta }_p(P(\varrho ,\varsigma ),P(\xi ,\tau ))& \precsim \lambda \otimes [{\delta }_p(\varrho ,\xi )\vee {\delta }_p(\varsigma ,\tau )]\hfill \\ \hfill & \precsim \lambda \otimes [\theta \vee {\delta }^{*}(\varsigma ,\tau )]\hfill \\ \hfill & =\lambda \otimes [{\delta }^s(\varrho ,\xi )\vee {\delta }^{*}(\varsigma ,\tau )].\hfill \end{array}$$
• Case III. Let $\varrho \ne \xi$ and $\varsigma =\tau$. Then, we obtain

  $$\begin{array}{cc}\hfill {\delta }^{*}(P(\varrho ,\varsigma ),P(\xi ,\tau ))={\delta }_p(P(\varrho ,\varsigma ),P(\xi ,\tau ))& \precsim \lambda \otimes [{\delta }_p(\varrho ,\xi )\vee {\delta }_p(\varsigma ,\tau )]\hfill \\ \hfill & \precsim \lambda \otimes [{\delta }^{*}(\varrho ,\xi )\vee \theta ]\hfill \\ \hfill & =\lambda \otimes [{\delta }^{*}(\varrho ,\xi )\vee {\delta }^s(\varsigma ,\tau )].\hfill \end{array}$$

Consequently, Theorem 1 ensures that P possesses exactly one coupled fixed point. □

As mentioned in the Preliminaries, QVQMSs can be seen as a generalization of PMSs. We present the following result, which serves as an application of Theorem 1 to PMSs.

Corollary 10. 

Consider a strong complete PMS $(S,\Psi ,\divideontimes )$, where ⋇ is assumed to be a left-continuous t-norm. Assume that the mapping $P:S\times S\to S$ satisfies, for all $\varrho ,\varsigma ,\xi ,\tau \in S$, the inequality

$${\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right)\ge max\{{\Psi }_{\varrho ,\xi }({\displaystyle \frac{t}{\lambda }}),{\Psi }_{\varsigma ,\tau }({\displaystyle \frac{t}{\lambda }})\},$$

(17)

where $t\in [0,\infty ),\lambda \in (0,1)$. Suppose, furthermore, that there exists a pair $({\varrho }_0,{\varsigma }_0)\in S\times S$, such that

$$\underset{t\to \infty }{lim}max\{{\Psi }_{{\varrho }_0,P({\varrho }_0,{\varsigma }_0)}\left(t\right),{\Psi }_{{\varsigma }_0,P({\varsigma }_0,{\varrho }_0)}\left(t\right)\}=1.$$

(18)

Then, P admits a unique coupled fixed point.

Proof. 

Consider the quantale action structure $(\Phi ,{\le }^{op},\oplus ,\otimes )$, where $(\Phi ,{\le }^{op},\oplus )$ is defined in the Preliminaries and ⊗ is given in (2). Then $(S,\Psi )$ forms an s-complete QVQMS. We now verify that condition (17) implies condition (3). Let us take $\varrho ,\varsigma ,\xi ,\tau \in S$ and $t\in [0,\infty ),\lambda \in (0,1)$, such that (17) holds. Under these assumptions, we obtain

$$\begin{array}{c}\hfill {\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right)\ge max\{(\lambda \otimes {\Psi }_{\varrho ,\xi })\left(t\right),(\lambda \otimes {\Psi }_{\varsigma ,\tau })\left(t\right)\}\end{array}$$

From the definition of the opposite order relation, we obtain

$$\begin{array}{cc}\hfill {\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}& {\le }^{op}max\{\lambda \otimes {\Psi }_{\varrho ,\xi },\lambda \otimes {\Psi }_{\varsigma ,\tau }\}\hfill \\ \hfill & =\lambda \otimes [max\{{\Psi }_{\varrho ,\xi },{\Psi }_{\varsigma ,\tau }\}].\hfill \end{array}$$

Furthermore, condition (18) ensures condition (4). Hence, by Theorem 1, the existence of a unique coupled fixed point is guaranteed. □

As an application of Corollary 1 in the setting of probabilistic metric spaces, the following result is obtained.

Corollary 11. 

Consider a strong complete PMS $(S,\Psi ,\divideontimes )$, where ⋇ is assumed to be a left-continuous t-norm. Assume that the mapping $P:S\times S\to S$ satisfies, for all $\varrho ,\varsigma ,\xi ,\tau \in S$, the inequality

$${\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right)\ge max\{{\Psi }_{\varrho ,\xi }({\displaystyle \frac{t}{\alpha }}),{\Psi }_{\varsigma ,\tau }({\displaystyle \frac{t}{\beta }})\},$$

(19)

where $t\in [0,\infty ),\alpha ,\beta >0$ with $\alpha +\beta <1$. Suppose, furthermore, that there exists a pair $({\varrho }_0,{\varsigma }_0)\in S\times S$, such that

$$\underset{t\to \infty }{lim}max\{{\Psi }_{{\varrho }_0,P({\varrho }_0,{\varsigma }_0)}\left(t\right),{\Psi }_{{\varsigma }_0,P({\varsigma }_0,{\varrho }_0)}\left(t\right)\}=1.$$

Then, P admits a unique coupled fixed point.

Proof. 

Consider the quantale action structure $(\Phi ,{\le }^{op},\oplus ,\otimes )$, where $(\Phi ,{\le }^{op},\oplus )$ is a value quantale, such that ⊗ is given in (2) and ⊕ is defined as follows:

$$({\Psi }_1\oplus {\Psi }_2)\left(x\right)=\underset{y+z=x}{\bigvee }min\{{\Psi }_1\left(y\right),{\Psi }_2\left(z\right)\}$$

for all ${\Psi }_1,{\Psi }_2\in \Phi$ and $x\in [0,\infty )$.

Then $(S,\Psi )$ forms an s-complete QVQMS. We now verify that condition (19) implies condition (5). Let us take $\varrho ,\varsigma ,\xi ,\tau \in S$ and $t\in [0,\infty ),\alpha ,\beta >0,\alpha +\beta <1$, such that (19) holds. Under these assumptions and by the definition of quantale operation ⊕, we obtain

$$\begin{array}{cc}\hfill {\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right)& \ge max\{(\alpha \otimes {\Psi }_{\varrho ,\xi })\left(t\right),(\beta \otimes {\Psi }_{\varsigma ,\tau })\left(t\right)\}\hfill \\ \hfill & \ge [(\alpha \otimes {\Psi }_{\varrho ,\xi })\oplus (\beta \otimes {\Psi }_{\varsigma ,\tau })]\left(t\right)\hfill \end{array}$$

From the definition of the opposite order relation, we obtain

$$\begin{array}{cc}\hfill {\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}& {\le }^{op}[(\alpha \otimes {\Psi }_{\varrho ,\xi })\oplus (\beta \otimes {\Psi }_{\varsigma ,\tau })].\hfill \end{array}$$

Hence, by the Corollary 1, the existence of a unique coupled fixed point is guaranteed. □

We conclude the paper with the following Corollary, which is a consequence of Theorem 2.

Corollary 12. 

Consider a strong complete PMS $(S,\Psi ,\divideontimes )$, where ⋇ is assumed to be a left-continuous t-norm. Assume that the mapping $P:S\times S\to S$ satisfies, for all $\varrho ,\varsigma ,\xi ,\tau \in S$, the inequality

$$max\{{\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right),{\Psi }_{P(\varsigma ,\varrho ),P(\tau ,\xi )}\left(t\right)\}\ge max\{{\Psi }_{\varrho ,\xi }({\displaystyle \frac{t}{\lambda }}),{\Psi }_{\varsigma ,\tau }({\displaystyle \frac{t}{\lambda }})\},$$

(20)

where $t\in [0,\infty ),\lambda \in (0,1)$. Suppose, furthermore, that there exists a pair $({\varrho }_0,{\varsigma }_0)\in S\times S$, such that

$$\underset{t\to \infty }{lim}max\{{\Psi }_{{\varrho }_0,P({\varrho }_0,{\varsigma }_0)}\left(t\right),{\Psi }_{{\varsigma }_0,P({\varsigma }_0,{\varrho }_0)}\left(t\right)\}=1.$$

Then P admits a unique coupled fixed point.

Proof. 

Consider the quantale action structure given in the proof of Corollary 10. It suffices to show that (20) implies (11). Let us take $\varrho ,\varsigma ,\xi ,\tau \in S$ and $t\in [0,\infty ),\lambda \in (0,1)$, such that (20) holds. Then, we obtain

$$\begin{array}{c}\hfill max\{{\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )}\left(t\right),{\Psi }_{P(\varsigma ,\varrho ),P(\tau ,\xi )}\left(t\right)\}\ge max\{(\lambda \otimes {\Psi }_{\varrho ,\xi })\left(t\right),(\lambda \otimes {\Psi }_{\varsigma ,\tau })\left(t\right)\}.\end{array}$$

By the definition of the opposite order relation, we obtain

$$\begin{array}{c}\hfill max\{{\Psi }_{P(\varrho ,\varsigma ),P(\xi ,\tau )},{\Psi }_{P(\varsigma ,\varrho ),P(\tau ,\xi )}\}{\le }^{op}\lambda \otimes [max\{{\Psi }_{\varrho ,\xi },{\Psi }_{\varsigma ,\tau }\}].\end{array}$$

Therefore, by Theorem 2, we obtain that P admits a unique coupled fixed point. □

5. Conclusions

Many generalizations of metric spaces have been obtained by modifying the classical metric axioms. For instance, by dropping the symmetry requirement, quasi-metric spaces were introduced [1]. Since many distance functions arising in real-life situations are inherently non-symmetric, quasi-metrics play a significant role in the mathematical modelling of various applied problems. Another important generalized metric structure, particularly relevant in theoretical computer science, is that of partial metric spaces [2]. Most of these generalizations have been achieved through suitable alterations of the standard metric axioms. In 1977, Flagg [4] introduced quantale-valued metric spaces by replacing the classical value set $[0,\infty ]$ with a value quantale. This structure provides an abstract algebraic model capturing the essential properties of the usual addition and usual order on the non-negative real numbers. Quantale-based frameworks have proven to be highly useful and flexible, especially in theoretical computer science. Indeed, many problems in computer science can be represented by functions assigning a solution to each problem instance; such functions are referred to as a solution operators (see [13]). In [13], Siedlecki employed quantale-valued metrics to introduce a generalized notion of solution operators, which has become an important tool in the study of computational problems modelled by the partially ordered sets. Motivated by these developments, in the present paper, we have extended coupled fixed point theorems from classical metric spaces to the broader setting of quantale-valued quasi-metric spaces, within the framework of abstract analysis and without resorting to categorical language. We have shown that, under appropriate conditions, coupled fixed point results can still be established in these highly abstract structures. The theoretical results obtained are supported by illustrative examples, and their applicability is demonstrated through applications to QVPMSs and PMSs. As a direction for future research, we plan to investigate fixed point theorems for solution operators in quantale-valued quasi-metric spaces. In particular, we aim to develop best proximity point results in this framework and to explore their potential applications in theoretical computer science.